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Theorem spesbcdi 32878
Description: A lemma for introducing an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
Hypotheses
Ref Expression
spesbcdi.1 (𝜑𝜓)
spesbcdi.2 ([𝐴 / 𝑥]𝜒𝜓)
Assertion
Ref Expression
spesbcdi (𝜑 → ∃𝑥𝜒)

Proof of Theorem spesbcdi
StepHypRef Expression
1 spesbcdi.1 . . 3 (𝜑𝜓)
2 spesbcdi.2 . . 3 ([𝐴 / 𝑥]𝜒𝜓)
31, 2sylibr 222 . 2 (𝜑[𝐴 / 𝑥]𝜒)
43spesbcd 3487 1 (𝜑 → ∃𝑥𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wex 1694  [wsbc 3401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-v 3174  df-sbc 3402
This theorem is referenced by: (None)
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